Method and Apparatus for Adaptive Parallel Proportional-Integral-Derivative Controller

ABSTRACT

An adaptive parallel proportional-integral-derivative controller produces a fixed controller output including a fixed proportional-integral-derivative and a fixed feedforward controller command, and an adaptive controller output including an adaptive parallel PID and an adaptive feedforward command all from a reference command. The fixed controller output and the adaptive controller output are added to produce a control command for a controlled system, which provides a measure of an output and a rate of change of the output as feedback for the controller.

RELATED APPLICATION

This U.S. patent application is related to U.S. patent application Ser.No. 12/057,721, co-filed herewith, and incorporated herein by reference.

FIELD OF THE INVENTION

This invention relates generally to controlled systems, in particular,to real-time adaptive proportional-integral-derivative (PID)controllers.

BACKGROUND OF THE INVENTION

Many controlled systems use proportional-integral-derivative (PID)controllers. However, a performance and stability of these controlledsystems is sensitive to system parameters, such as inertia or stiffnessand selected PID gains.

Adaptive control is one possible method for improving the performance ofthese controlled systems. However, adaptive control usually requiresdetailed process models or an approximation of these models to estimatethe system parameters.

For example, U.S. Pat. No. 5,444,612 and U.S. Pat. No. 5,691,615describe motion controllers with adaptive control based on a motionsystem model to estimate and compensate for inertia, damping, friction,and gravity parameters and perform model-based adaptive control. U.S.Pat. No. 6,055,524 and U.S. Pat. No. 5,687,077 describe functionapproximation methods, such as neural networks and Laguerre functions,to approximate the system model and estimate the correspondingparameters. Other approaches, such as U.S. Pat. No. 6,658,370, andreferences therein, describe some type of adaptive control by using afinite set of pre-designed sets of tuning constants, and a method todetermine which set of tuning constants are optimum. That approachrequires that at least one set of pre-designed tuning constants yieldsacceptable performance for an unknown system in operation. Other typesof PID controllers use rule based adjustment of controller gains, suchas fuzzy logic conditions.

Related approaches are described for adaptive parallel PID by Chang,W.-D., and J.-J Yan, “Adaptive robust PID controller design based on asliding inode for uncertain chaotic systems,” Chaos, Solitons andFractals, 26, pp. 167-175, 2005, Iwai, Z., Mizumoto, I., Liu, L.; Shah,S. L.; Jiang, H., “Adaptive Stable PID Controller with ParallelFeedforward Compensator,” Conference on Control, Automation, Roboticsand Vision, December, 2006, Pirabakaran, K., and V. M. Bacerra,“Automatic Tuning of PID Controllers Using Model Reference AdaptiveControl Techniques,” Conference of the IEEE Industrial ElectronicsSociety, December, 2001, and Xiong, A. and Y. Fan, “Application of a PIDController using MRAC Techniques for Control of the DC ElectromotorDrive,” IEEE International Conference on Mechatronics and Automation,August, 2007.

SUMMARY OF THE INVENTION

An adaptive parallel proportional-integral-derivative controllerproduces a fixed controller output including a fixedproportional-integral-derivative and a fixed feedforward controllercommand, and an adaptive controller output including an adaptiveparallel PID and an adaptive feedforward command all from a referencecommand.

The fixed controller output and the adaptive controller output are addedto produce a control command for a controlled system, which provides ameasure of an output and a rate of change of the output as feedback forthe controller.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an adaptive PID controller according to anembodiment of the invention;

FIG. 2 is a block diagram of an adaptive parallel PID controlleraccording to an embodiment of the invention;

FIG. 3 is a block diagram of an adaptive parallel PID controller withoverall proportional gain according to an embodiment of the invention;

FIG. 4 is a block diagram of an adaptive parallel PID controller withoverall derivative gain according to an embodiment of the invention; and

FIG. 5 is a block diagram of an adaptive parallel PID controller withoverall integral gain according to an embodiment of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

As shown in FIG. 1, the embodiments of the invention provide anapparatus and method 100 for adaptive proportional-integral-derivative(PID) control with feedforward control. The method and apparatuscomprise a fixed controller 110 and an adaptive controller 120. Thefixed controller includes a fixed PID controller, and a fixedfeedforward controller. The adaptive controller includes an adaptiveparallel controller 120 and an adaptive feedforward controller.

Input to the fixed and adaptive controllers is a reference command r101. An output of the fixed controller is a fixed PID and a fixedfeedforward 111. An output 121 of the adaptive parallel controller 120is an adaptive parallel PID 123 and an adaptive feedforward 122, seeFIG. 2-4, which in combination form an adaptive parallel control commandu_(adapt) 121.

The fixed output 111 and the adaptive output are added 130 to form acontrol command u 131 for a controlled system 140. The controlled systemcan provide a measure of an output y 141 and a rate of change 142 of theoutput 141, using some sensing or approximation means 143, which are fedback to the fixed and adaptive parallel controllers. The controlledsystem can be any system as known in the art.

The controlled system 140 can be of dominant order n, with n=1 for firstorder dominant processes, such as most temperature or velocitycontrolled systems and n=2 for second order dominant processes, such asmost position controlled systems. In the preferred embodiment, n≦2,because most controlled systems have dominant first and second orderdynamics.

The following equalities are defined:

z=−(d/dt+K _(pp))^(n−1) e

z _(I) =∫zdt

where t is time, K_(pp)>0 is a selected scalar gain, e=r−y is a trackingerror for the reference command r 101, and y is the output.

The control command u 131 for the preferred embodiment is

u=K _(pv) z+K _(iv) z _(I) +K _(ff) w _(ff) +u _(adapt)  (1a)

u _(adapt) ={circumflex over (K)} _(ff) w _(ff) +u _(a)  (1b)

where K_(pv)>0 is a fixed proportional gain, K_(iv)>0 is a fixedintegral gain, K_(ff) is a fixed feedforward gain. The adaptive parallelcontrol command u_(adapt) 121 includes an adaptive parallel PID commandu_(a) and an adaptive feedforward command {circumflex over(K)}_(ff)w_(ff). The fixed and adaptive feedforward gains K_(ff) and{circumflex over (K)}_(ff), respectively are multiplied by a combinedfeedforward and feedback signal w_(ff)=y^((n))−ż.

For n=1, the fixed controller is a PI controller, and signal w_(ff)={dotover (r)}. For n=2, the fixed controller is a PID controller, and thesignal w_(ff)={umlaut over (r)}+K_(pp)ė. The (.) and (..) superscriptsof the variables denote first and second derivatives with respect totime.

The adaptive feedforward gain {circumflex over (K)}_(ff) is updatedaccording to:

{circumflex over ({dot over (K)} _(ff)=−γ_(ff) w _(ff) z−L _(ff){circumflex over (K)} _(ff),  (2)

where γ_(ff)>0 is an adaptation gain for the adaptive feedforward gain,and L_(ff)≧0 is a filter gain.

The adaptive parallel PID control command u_(a) and a method forupdating the adaptive gains depends on the embodiment of the adaptiveparallel PID controller, as described below for different parallel PIDembodiments.

Parallel PID Controller

FIG. 2 shows the adaptive parallel PID controller 120 in greater detail.The adaptive parallel PID command u_(a) 123 is

u _(a) ={circumflex over (K)} _(P) e+{circumflex over (K)} _(D)ė+{circumflex over (K)} _(I) ∫edt,  (3)

where {circumflex over (K)}_(P) 210 is an adaptive proportional gain,{circumflex over (K)}_(D) 220 is an adaptive derivative gain,{circumflex over (K)}_(I) 230 is an adaptive integral gain. The module250 is a differentiator, in which s is a Laplace variable.

The adaptive gains for Equation (3) are updated according to

{circumflex over ({dot over (K)} _(P)=−γ_(P) ez−L _(P) {circumflex over(K)} _(P)  (4)

{circumflex over ({dot over (K)} _(D)=−γ_(D) ėz−L _(D) {circumflex over(K)} _(D)  (5)

{circumflex over ({dot over (K)} _(I)=−γ_(I) ∫edtz−L _(I) {circumflexover (K)} _(I),  (6)

where γ_(P), γ_(D), γ_(I)>0 are adaptation gains for proportional,derivative, and integral gains, respectively. In addition, L_(P), L_(D),L_(I)≧0 are the filter gains used to adjust adaptation response for theadaptive PID gains. Therefore, the overall design of the adaptiveparallel PID and feedforward controller 100 is given by Equations(1)-(6) in this embodiment with the adaptive parallel PID controller asshown in FIG. 2.

FIG. 3 shows an adaptive parallel controller 300 with overallproportional gain. In this embodiment, the adaptive parallel PID controlcommand is:

u _(a) ={circumflex over (K)} _(P)(e+{circumflex over (K)} _(Dp)ė+{circumflex over (K)} _(Ip) ∫edt)  (7)

where {circumflex over (K)}_(P) 210 is the adaptive proportional gain,{circumflex over (K)}_(Dp) 320 is an adaptive derivative gain scaled bythe adaptive proportional gain, K_(Ip) 330 is an adaptive integral gainscaled by the adaptive proportional gain.

The adaptive gains for the adaptive PID of Equation (7) are updatedaccording to

$\begin{matrix}{{\overset{.}{\hat{K}}}_{P} = {{{- {\gamma_{P}\left( {e + {\frac{{\hat{K}}_{D_{p}}}{2}\overset{.}{e}} + {\frac{{\hat{K}}_{I_{p}}}{2}{\int{e{t}}}}} \right)}}z} - {L_{P}{\hat{K}}_{P}}}} & (8) \\{{\overset{.}{\hat{K}}}_{D_{p}} = {{{- \gamma_{D}}\frac{{\hat{K}}_{P}}{2}\overset{.}{e}z} - {L_{D_{p}}{\hat{K}}_{D_{p}}}}} & (9) \\{{{\overset{.}{\hat{K}}}_{I}}_{p} = {{{- \gamma_{I}}\frac{{\hat{K}}_{P}}{2}{\int{e{{tz}}}}} - {L_{I_{p}}{\hat{K}}_{I_{p}}}}} & (10)\end{matrix}$

where γ_(P), γ_(Dp), γ_(Ip)>0 are the adaptation gains for proportional,derivative, and integral gains, respectively. In addition, L_(P),L_(Dp), L_(Ip)≧ are the filter gains used to an adjust adaptationresponse for the adaptive PID gains. Therefore, the overall design ofthe adaptive parallel PID and feedforward controller 100 is given byEquations (1), (2), (7) and, (8)-(10) in this embodiment with theadaptive parallel PID controller according to FIG. 3.

Parallel PID Controller with Overall Derivative Gain

FIG. 4 shows an adaptive parallel controller 400 with overall derivativegain. In this embodiment, the adaptive parallel PID control command is

u _(a) ={circumflex over (K)} _(D)(ė+{circumflex over (K)} _(Pd)e+{circumflex over (K)} _(Id) ∫edt),  (11)

where {circumflex over (K)}_(D) 220 is the adaptive derivative gain,{circumflex over (K)}_(Pd) 410 is an adaptive proportional gain scaledby the adaptive the derivative gain, and {circumflex over (K)}_(Id) 420is an adaptive integral gain scaled by the adaptive derivative gain.

The adaptive gains for the adaptive PID of Equation (11) are updatedaccording to

$\begin{matrix}{{\overset{.}{\hat{K}}}_{Pd} = {{{- \gamma_{Pd}}\frac{{\hat{K}}_{D}}{2}{ez}} - {L_{Pd}{\hat{K}}_{Pd}}}} & (12) \\{{\overset{.}{\hat{K}}}_{D} = {{{- {\gamma_{D}\left( {\overset{.}{e} + {\frac{{\hat{K}}_{Pd}}{2}e} + {\frac{{\hat{K}}_{Id}}{2}{\int{e{t}}}}} \right)}}z} - {L_{D}{\hat{K}}_{D}}}} & (13) \\{{\overset{.}{\hat{K}}}_{Id} = {{{- \gamma_{I}}\frac{{\hat{K}}_{D}}{2}{\int{e{{tz}}}}} - {L_{Id}{\hat{K}}_{Id}}}} & (14)\end{matrix}$

where γ_(Pd), γ_(D), γ_(Id)>0 are adaptation gains for proportional,derivative, and integral gains, respectively. In addition, L_(Pd),L_(D), L_(Id)≧ are filter gains used to adjust adaptation response foradaptive PID gains. Therefore, the overall design of the adaptiveparallel PID and feedforward controller 100 is given by Equations (1),(2), (11) and, (12)-(14) in this embodiment with the adaptive parallelPID controller according to FIG. 4.

Parallel PID Controller with Overall Integral Gain

FIG. 5 shows an adaptive parallel controller 500 with overall integralgain. In this embodiment, the adaptive parallel PID control command is

u _(a) ={circumflex over (K)} _(I)(∫edt+{circumflex over (K)} _(Pi)e+{circumflex over (K)} _(Di) ė)  (15)

where {circumflex over (K)}_(I) 230 is an adaptive integral gain,{circumflex over (K)}_(Di) 520 is an adaptive derivative gain scaled bythe adaptive integral gain, {circumflex over (K)}_(Pi) 510 is anadaptive proportional gain scaled by the adaptive integral gain. Theadaptive gains for the adaptive PID of Equation (15) are updatedaccording to

$\begin{matrix}{{\overset{.}{\hat{K}}}_{Pi} = {{{- \gamma_{Pi}}\frac{{\hat{K}}_{I}}{2}{ez}} - {L_{Pi}{\hat{K}}_{Pi}}}} & (16) \\{{\overset{.}{\hat{K}}}_{Di} = {{{- \gamma_{Di}}\frac{{\hat{K}}_{I}}{2}\overset{.}{e}z} - {L_{Di}{\hat{K}}_{Di}}}} & (17) \\{{\overset{.}{\hat{K}}}_{I} = {{{- {\gamma_{I}\left( {{\int{e{t}}} + {\frac{{\hat{K}}_{Pi}}{2}e} + {\frac{{\hat{K}}_{Di}}{2}\overset{.}{e}}} \right)}}z} - {L_{I}{\hat{K}}_{I}}}} & (18)\end{matrix}$

where γ_(P), γ_(D), γ_(I)>0 are adaptation gains for proportional,derivative, and integral gains, respectively. In addition, L_(Pi),L_(Di), L_(I)≧ are filter gains used to adjust adaptation response forthe adaptive gains. Therefore, the overall design of the adaptiveparallel PID and feedforward controller 100 is given by Equations (1),(2), (15) and, (16)-(18) in this embodiment with the adaptive parallelPID controller according to FIG. 5.

Design Considerations

For a class of controlled systems:

ay ^((n)) =u,  (19)

where y^((n)) is the n_(th) derivative of the output y, where n is adominant order of the system. An unknown constant parameter a>0 is ahigh frequency gain. The Equations (1)-(2) are substituted into Equation(19), which yields

aż=−K _(pu) z−K _(iu) z _(I) +{tilde over (K)} _(ff) w _(ff) +u _(a)

where

{tilde over (K)} _(ff) ={circumflex over (K)} _(ff) −a+K _(ff)

is a feedforward gain estimation error. Consider the Lyapunov potentialfunction

V=az ² +K _(iu) z _(I) ²+γ_(ff) ⁻¹ {tilde over (K)} _(ff) ² +{circumflexover (K)}′Γ ⁻¹ {circumflex over (K)},

where {circumflex over (K)} is a three element vector including thethree adaptive PID gains for any of the embodiments described above, andΓ is the diagonal adaptation gain matrix including the adaptation gains,such as γ_(P), γ_(D), and γ_(I) for the embodiment according to Equation(3), and so on.

The design of the adaptive controller is based on obtaining negativityof the function derivative {dot over (V)}. Determining {dot over (V)}yields

{dot over (V)}=−K _(pu) z ² +zu _(a) +{circumflex over (K)}′Γ ⁻¹{circumflex over ({dot over (K)}.

Using the formula for the adaptive PID control command in Equation (3),and the corresponding adaptation Equations (4)-(6) for {dot over (V)}≦0,and substituting into the above equation for {dot over (V)} shows that{dot over (V)}≦0 and thus proves system stability.

The same procedure can be repeated for the three other adaptive PIDembodiments in Equations (7), (11), and (15), and their correspondingadaptation Equations to prove stability of the system with {dot over(V)}≦0 according to the Lyapunov stability theory. The adaptationEquations used to update the gains are obtained using the approachdescribed above.

In particular, a general formulation for updating the adaptation gainvector {circumflex over (K)} including adaptive PID gains associatedwith the adaptive PID command u_(a), is

${{u_{a}\left( {\hat{K},t} \right)} = {{u_{a}\left( {0,t} \right)} + {{\hat{K}}^{\prime}{\nabla{u_{a}\left( {0,t} \right)}}} + {\frac{1}{2}{\hat{K}}^{\prime}{\nabla^{2}u_{a}}\hat{K}}}},$

where u_(a)({circumflex over (K)}, e, ė, ∫e) is denoted byu_(a)({circumflex over (K)}, t), and ∇u_(a)(0, t) is the gradient, i.e.,the first order derivative, of the adaptive PID control command u_(a)with respect to {circumflex over (K)} and evaluated at {circumflex over(K)}=0. ∇²u_(a) is the Hessian or second order derivative of u_(a) withrespect to {circumflex over (K)}, which is independent of {circumflexover (K)} in this case. The above formula is obtained by using an exactsecond order Taylor expansion of u_(a).

EFFECT OF THE INVENTION

The invention provides adaptive PID controller and method fordynamically adjusting adaptive PID gains for parallel PID with coupledgain adaptation using output and output rate feedback signals and areference command. The invention can operate without using detailedprocess models or their approximation for parameter estimation, orpredetermined gain values as in most conventional PID controllers. Theembodiments of the invention can use an overall proportional gain oroverall integral gain. The adaptive PID controller can be used tocompensate for possibly unknown and varying system parameters such asstiffness and inertia of a controlled system.

Although the invention has been described with reference to certainpreferred embodiments, it is to be understood that various otheradaptations and modifications can be made within the spirit and scope ofthe invention. Therefore, it is the object of the append claims to coverall such variations and modifications as come within the true spirit andscope of the invention.

1. An apparatus for controlling a system, comprising: a fixed controllercomprising: a fixed proportional-integral-derivative (PID) controller;and a fixed feedforward controller, in which the fixed PID controllerand the fixed feedforward controller are configured to receive areference command and to produce a fixed controller output comprising afixed PID and a fixed feedforward controller command; an adaptiveparallel controller comprising: an adaptive parallel PID controller; andan adaptive feedforward controller, in which the adaptive parallel PIDcontroller and the adaptive feedforward controller are configured toreceive the reference command and to produce adaptive controller outputcomprising an adaptive parallel PID and an adaptive feedforward; andmeans for adding the fixed controller output and the adaptive parallelcontroller output to produce a control command for a controlled system,the controlled system providing a measure of an output and a rate ofchange of the output as feedback to the fixed controller and theadaptive controller.
 2. The apparatus of claim 1, in which gains of theadaptive parallel controller and the adaptive feedforward controller areadjusted dynamically.
 3. The apparatus of claim 1, in which thecontrolled system has a dominant order n, with n=1 for first orderdominant processes, such as temperature or velocity controlled systems.4. The apparatus of claim 1, in which the controlled system has adominant order n, with n=2 for second order dominant processes, such asposition controlled systems.
 5. The apparatus of claim 1, in whichz=−(d/dt+K _(pp))^(n−1) ez _(I) =∫zdt where t is time, K_(pp)>0 is a selected scalar gain, e=r−yis a tracking error for the reference command r, and y is the output. 6.The apparatus of claim 5, in which the control command isu=K _(pu) z+K _(iu) z _(I) +K _(ff) w _(ff) +u _(adapt)u _(adapt) ={circumflex over (K)} _(ff) w _(ff) +u _(a) where K_(pv)>0is a fixed proportional gain, K_(iv)>0 is a fixed integral gain, K_(ff)is a fixed feedforward gain, and an adaptive parallel control commandu_(adapt) includes an adaptive parallel PID command u_(a) and anadaptive feedforward command {circumflex over (K)}_(ff)w_(ff), and thefixed feedforward gain K_(ff) and the adaptive feedforward gain{circumflex over (K)}_(ff) are multiplied by a combined feedforward andfeedback signal w_(ff)=y^((n))−ż.
 7. The apparatus of claim 6, in whichthe controlled system has a dominant order n=1 for first order dominantprocesses, and the fixed controller is a PI controller, and w_(ff)={dotover (r)}, where a superscript (.) on the variable r denote a firstderivative with respect to time t.
 8. The apparatus of claim 6, in whichthe controlled system has a dominant order n=2 for second order dominantprocesses, and the fixed controller is a PID controller, andw_(ff)={umlaut over (r)}+K_(pp)ė, where a superscript (..) on thevariable r denote a second derivative with respect to time t.
 9. Theapparatus of claim 6, in which the adaptive feedforward gain {circumflexover (K)}_(ff) is updated according to:{circumflex over ({dot over (K)} _(ff)=−γ_(ff) w _(ff) z−L _(ff){circumflex over (K)} _(ff), where γ_(ff)>0 is an adaptation gain forthe adaptive feedforward gain, and L_(ff)≧0 is a filter gain.
 10. Theapparatus of claim 6, in which the adaptive parallel PID control commandu_(a) isu _(a) ={circumflex over (K)} _(P) e+{circumflex over (K)} _(D)ė+{circumflex over (K)} _(I) ∫edt, where {circumflex over (K)}_(P) is anadaptive proportional gain, {circumflex over (K)}_(D) is an adaptivederivative gain, {circumflex over (K)}_(I) is an adaptive integral gain.The module 250 is a differentiator, in which s is a Laplace variable.11. The apparatus of claim 10, in which adaptive gains of the adaptiveparallel PID control command are updated according to{circumflex over ({dot over (K)} _(P)=−γ_(P) ez−L _(P) {circumflex over(K)} _(P){circumflex over ({dot over (K)} _(D)=−γ_(D) ėz−L _(D) {circumflex over(K)} _(D){circumflex over ({dot over (K)} _(I)=−γ_(I) ∫edtz−L _(I) {circumflexover (K)} _(I), where γ_(P), γ_(D), γ_(I)>0 are adaptation gains forproportional, derivative, and integral gains, respectively and L_(P),L_(D), L_(I)≧0 are filter gains used to adjust an adaptation responsefor adaptive PID gains.
 12. The apparatus of claim 6, in which theadaptive parallel PID control command isu _(a) ={circumflex over (K)}(e+{circumflex over (K)} _(Dp)ė+{circumflex over (K)} _(IP) ∫edt), where {circumflex over (K)}_(P) isan adaptive proportional gain, {circumflex over (K)}_(Dp) is an adaptivederivative gain scaled by the adaptive proportional gain, K_(Ip) is anadaptive integral gain scaled by the adaptive proportional gain.
 13. Theapparatus of claim 12, in which adaptive gains of the adaptive parallelPID control command are updated according to $\begin{matrix}{{\overset{.}{\hat{K}}}_{P} = {{{- {\gamma_{P}\left( {e + {\frac{{\hat{K}}_{D_{p}}}{2}\overset{.}{e}} + {\frac{{\hat{K}}_{I_{p}}}{2}{\int{e{t}}}}} \right)}}z} - {L_{P}{\hat{K}}_{P}}}} \\{{\overset{.}{\hat{K}}}_{D_{p}} = {{{- \gamma_{D}}\frac{{\hat{K}}_{P}}{2}\overset{.}{e}z} - {L_{D_{p}}{\hat{K}}_{D_{p}}}}} \\{{{{\overset{.}{\hat{K}}}_{I}}_{p} = {{{- \gamma_{I}}\frac{{\hat{K}}_{P}}{2}{\int{e{{tz}}}}} - {L_{I_{p}}{\hat{K}}_{I_{p}}}}},}\end{matrix}$ where γ_(P), γ_(Dp), γ_(Ip)>0 are adaptation gains forproportional, derivative, and integral gains, respectively and L_(P),L_(Dp), L_(Ip)≧ are the filter gains used to adjust an adaptationresponse for the adaptive PID gains.
 14. The apparatus of claim 6, inwhich the adaptive parallel PID control command isu _(a) ={circumflex over (K)} _(D)(ė+{circumflex over (K)} _(Pd)e+{circumflex over (K)} _(Id) ∫edt), where {circumflex over (K)}_(D) isan adaptive derivative gain, {circumflex over (K)}_(Pd) is an adaptiveproportional gain scaled by the adaptive the derivative gain, and{circumflex over (K)}_(Id) 420 is an adaptive integral gain scaled bythe adaptive derivative gain.
 15. The apparatus of claim 14, in whichthe adaptive gains are updated according to $\begin{matrix}{{\overset{.}{\hat{K}}}_{Pd} = {{{- \gamma_{Pd}}\frac{{\hat{K}}_{D}}{2}{ez}} - {L_{Pd}{\hat{K}}_{Pd}}}} \\{{\overset{.}{\hat{K}}}_{D} = {{{- {\gamma_{D}\left( {\overset{.}{e} + {\frac{{\hat{K}}_{Pd}}{2}e} + {\frac{{\hat{K}}_{Id}}{2}{\int{e{t}}}}} \right)}}z} - {L_{D}{\hat{K}}_{D}}}} \\{{{\overset{.}{\hat{K}}}_{Id} = {{{- \gamma_{I}}\frac{{\hat{K}}_{D}}{2}{\int{e{{tz}}}}} - {L_{Id}{\hat{K}}_{Id}}}},}\end{matrix}$ where γ_(Pd), γ_(D), γ_(Id)>0 are adaptation gains forproportional, derivative, and integral gains, respectively and L_(Pd),L_(D), L_(Id)≧ are filter gains used to adjust an adaptation responsefor the adaptive gains.
 16. The apparatus of claim 6, in which theadaptive parallel PID control command isu _(a) ={circumflex over (K)} _(I)(∫edt+{circumflex over (K)} _(Pi)e+{circumflex over (K)} _(Di) ė), where {circumflex over (K)}_(I) is anadaptive integral gain, {circumflex over (K)}_(Di) is an adaptivederivative gain scaled by the adaptive integral gain, {circumflex over(K)}_(Pi) is an adaptive proportional gain scaled by the adaptiveintegral gain.
 17. The apparatus of claim 16, in which the adaptivegains are updated according to $\begin{matrix}{{\overset{.}{\hat{K}}}_{Pi} = {{{- \gamma_{Pi}}\frac{{\hat{K}}_{I}}{2}{ez}} - {L_{Pi}{\hat{K}}_{Pi}}}} \\{{\overset{.}{\hat{K}}}_{Di} = {{{- \gamma_{Di}}\frac{{\hat{K}}_{I}}{2}\overset{.}{e}z} - {L_{Di}{\hat{K}}_{Di}}}} \\{{{\overset{.}{\hat{K}}}_{I} = {{{- {\gamma_{I}\left( {{\int{e{t}}} + {\frac{{\hat{K}}_{Pi}}{2}e} + {\frac{{\hat{K}}_{Di}}{2}\overset{.}{e}}} \right)}}z} - {L_{I}{\hat{K}}_{I}}}},}\end{matrix}$ where γ_(P), γ_(D), γ_(I)>0 are adaptation gains forproportional, derivative, and integral gains, respectively and L_(Pi),L_(Di), L_(I)≧ are filter gains used to adjust an adaptation responsefor the adaptive gains.
 18. The apparatus of claim 1, in which theadaptive parallel controller compensates for unknown and varying systemparameters such as stiffness and inertia of the controlled system.
 19. Amethod for controlling a system, comprising the steps of: producing afixed controller output comprising a fixedproportional-integral-derivative (PID) a fixed feedforward controllercommand from a reference command; producing an adaptive controlleroutput comprising an adaptive parallel PID and an adaptive feedforwardcommand from the reference command; and adding the fixed controlleroutput and the adaptive controller output to produce a control commandfor a controlled system providing a measure of an output and a rate ofchange of the output as feedback for the producing steps.
 20. The methodof claim 19, in which the adaptive controller output is produced inparallel.